Orbital stability vs. scattering in the cubic-quintic Schrodinger equation

Rémi Carles (1) , Christof Sparber (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 UIC - Department of Mathematics, Statistics and Computer Science [Chicago]

We consider the cubic-quintic nonlinear Schrödinger equation in space dimension up to three. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is paint a more or less complete picture of dispersion and orbital (in-)stability of solitary waves, emanating from nonlinear ground states. In space dimension one, it is already known that solitons are orbitally stable. Here, we establish the analogous result in dimension two. In addition, we show that if the initial data have at most the mass of the ground state for the cubic two-dimensional Schrödinger equation, then the solution is dispersive and asymptotically linear. Finally, in dimension three, relying on some previous results from other authors, we show that solitons may or may not be orbitally stable.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Rémi Carles, Christof Sparber. Orbital stability vs. scattering in the cubic-quintic Schrodinger equation. 2020. ⟨hal-02475123v1⟩